Geometric properties of generalized symmetric spaces

Abstract

Abstract. It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.

Ricci solitons on pseudo Riemannian generalized symmetric spaces

“…Deriving f 4 from (3) with respect to x 4 , then replacing into (11) and by taking (10) and ( 14) into account; we have…”

“…Cerny and Kowalski proved in [13] that four-dimensional proper (that is, non-symmetric) pseudo-Riemannian generalized symmetric spaces may be identified with R 4 equipped with a particular metric and are classified into four classes, named A, B, C and D, and the pseudo-Riemannian metrics can have any signature: the metric of type A is either Riemannian or neutral of signature (2,2), the metric of type C is Lorentzian and the metrics of type B and D are of signature (2,2). In [14]; Kowalski has proved the existence of generalized symmetric Riemannian spaces of arbitrary order, in [8] Calvaruso and De Leo have studied their curvature properties on the algebraic side using the Lie algebras, with respect to suitable pseudo-orthonormal bases and in [11], the authors showed that these spaces can naturally be equipped with some structures defined by their curvature tensors used to characterize symmetric spaces, as the existence of almost Hermitian and almost para-Hermitian structures. A complete classification up to isometry, of non-symmetric simply-connected fourdimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons was been given in [4]; where unlike type B, only types A, C and D are algebraic Ricci solitons.…”